10 CONVOLUTION OF TWO FUNCTIONS 291 function (5-, x)^g(x'1s)f(x) = g(5'1x)f(x) is (JJL ® 0)-integrable, and the right-hand side of (14.10.9.1) Is then equal to (14.10.9.2) Similarly, condition (b) means that the function (s9 x)t-+g(x)f(sx) is 0 integrable, and the left-hand side of (14.10.9.1) is then equal to (14.1 0.9.3) II g(x)f(sx) dn(s) df(x). 11 But from the theorem of Lebesgue-Fubini and the left-invariance of /?, it follows that one of the integrals (14.10.9.2), (14.10.9.3) exists if and only if the other one exists, and then both are equal. PROBLEMS 1. (a) Let g be a j3-measurable function of compact support on G. Show that / and g are convolvable and that/*# is continuous on G in the following two cases: (i) g is essentially bounded and / is locally /3-integrable; (ii) g e J^(G, ]8), /is jS-measurable and/2 is locally jS-integrable. (b) Assume that G is unimodular. Show that if /e ^(G, ]8) and g e J*?J(G, ]8), where p^\9q^\ and - 4- - ^ 1, then / and g are convolvable, f*ge J2?c(G, ]8) p % where - = - -f -— 1, and Nr(/* g) • :§ Na(/)N/#) (W. Young's inequality). (Consider r p q first the case where - + - = 1, then use (14,10.6.1) and the Riesz-Thorin theorem P Q (Section 13.17, Problem?).) 2. Let G be a compact group, ]8 the Haar measure on G for which j8(G) == 1, and let A, B be two /3-integrable sets. Show that for each e > 0 there exists x e G such that )3(A n xB) g (1 -f c))3(A)]5(B). Deduce that, for each integer H such that ]8(A) ^ l//i, there exist 71 points xt, ..,, xn in G such that jS(xiA u x2 A u • • • u xfj A) ^ ^. (Consider the sets .*/ • CA.) 3. Let G be a compact commutative group, ]8 the Haar measure on G for which /3(G) = 1, and g a /3-measurable function on G. Let (rn) be the orthonormal system of Rademacher functions (Section 13.21, Problem 10). Let a > 0 and let A be the set of x e G such that |000| >oc. Let n be an integer J> 1 such that n • /3(A) J> 1. Show that there exists a j3-measur- able set B such that j3(B) £> |, and « points si,...,sninG such that, if we put F(x, t) — n V rk(t)g(skx\ the following is true: for each x e B, there exists a finite union 100 of fc=i intervals in [0,1] such that (i) A(I(x)) ^ J (A being Lebesgue measure); (ii) for allh the